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| Mirrors > Home > MPE Home > Th. List > bcn1 | Structured version Visualization version GIF version | ||
| Description: Binomial coefficient: 𝑁 choose 1. (Contributed by NM, 21-Jun-2005.) (Revised by Mario Carneiro, 8-Nov-2013.) |
| Ref | Expression |
|---|---|
| bcn1 | ⊢ (𝑁 ∈ ℕ0 → (𝑁C1) = 𝑁) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elnn0 12528 | . 2 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) | |
| 2 | 1eluzge0 12934 | . . . . . . 7 ⊢ 1 ∈ (ℤ≥‘0) | |
| 3 | 2 | a1i 11 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → 1 ∈ (ℤ≥‘0)) |
| 4 | elnnuz 12922 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (ℤ≥‘1)) | |
| 5 | 4 | biimpi 216 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ (ℤ≥‘1)) |
| 6 | elfzuzb 13558 | . . . . . 6 ⊢ (1 ∈ (0...𝑁) ↔ (1 ∈ (ℤ≥‘0) ∧ 𝑁 ∈ (ℤ≥‘1))) | |
| 7 | 3, 5, 6 | sylanbrc 583 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 1 ∈ (0...𝑁)) |
| 8 | bcval2 14344 | . . . . 5 ⊢ (1 ∈ (0...𝑁) → (𝑁C1) = ((!‘𝑁) / ((!‘(𝑁 − 1)) · (!‘1)))) | |
| 9 | 7, 8 | syl 17 | . . . 4 ⊢ (𝑁 ∈ ℕ → (𝑁C1) = ((!‘𝑁) / ((!‘(𝑁 − 1)) · (!‘1)))) |
| 10 | facnn2 14321 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (!‘𝑁) = ((!‘(𝑁 − 1)) · 𝑁)) | |
| 11 | fac1 14316 | . . . . . . 7 ⊢ (!‘1) = 1 | |
| 12 | 11 | oveq2i 7442 | . . . . . 6 ⊢ ((!‘(𝑁 − 1)) · (!‘1)) = ((!‘(𝑁 − 1)) · 1) |
| 13 | nnm1nn0 12567 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0) | |
| 14 | 13 | faccld 14323 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (!‘(𝑁 − 1)) ∈ ℕ) |
| 15 | 14 | nncnd 12282 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → (!‘(𝑁 − 1)) ∈ ℂ) |
| 16 | 15 | mulridd 11278 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → ((!‘(𝑁 − 1)) · 1) = (!‘(𝑁 − 1))) |
| 17 | 12, 16 | eqtrid 2789 | . . . . 5 ⊢ (𝑁 ∈ ℕ → ((!‘(𝑁 − 1)) · (!‘1)) = (!‘(𝑁 − 1))) |
| 18 | 10, 17 | oveq12d 7449 | . . . 4 ⊢ (𝑁 ∈ ℕ → ((!‘𝑁) / ((!‘(𝑁 − 1)) · (!‘1))) = (((!‘(𝑁 − 1)) · 𝑁) / (!‘(𝑁 − 1)))) |
| 19 | nncn 12274 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ) | |
| 20 | 14 | nnne0d 12316 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (!‘(𝑁 − 1)) ≠ 0) |
| 21 | 19, 15, 20 | divcan3d 12048 | . . . 4 ⊢ (𝑁 ∈ ℕ → (((!‘(𝑁 − 1)) · 𝑁) / (!‘(𝑁 − 1))) = 𝑁) |
| 22 | 9, 18, 21 | 3eqtrd 2781 | . . 3 ⊢ (𝑁 ∈ ℕ → (𝑁C1) = 𝑁) |
| 23 | 0nn0 12541 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
| 24 | 1z 12647 | . . . . 5 ⊢ 1 ∈ ℤ | |
| 25 | 0lt1 11785 | . . . . . 6 ⊢ 0 < 1 | |
| 26 | 25 | olci 867 | . . . . 5 ⊢ (1 < 0 ∨ 0 < 1) |
| 27 | bcval4 14346 | . . . . 5 ⊢ ((0 ∈ ℕ0 ∧ 1 ∈ ℤ ∧ (1 < 0 ∨ 0 < 1)) → (0C1) = 0) | |
| 28 | 23, 24, 26, 27 | mp3an 1463 | . . . 4 ⊢ (0C1) = 0 |
| 29 | oveq1 7438 | . . . . 5 ⊢ (𝑁 = 0 → (𝑁C1) = (0C1)) | |
| 30 | eqeq12 2754 | . . . . 5 ⊢ (((𝑁C1) = (0C1) ∧ 𝑁 = 0) → ((𝑁C1) = 𝑁 ↔ (0C1) = 0)) | |
| 31 | 29, 30 | mpancom 688 | . . . 4 ⊢ (𝑁 = 0 → ((𝑁C1) = 𝑁 ↔ (0C1) = 0)) |
| 32 | 28, 31 | mpbiri 258 | . . 3 ⊢ (𝑁 = 0 → (𝑁C1) = 𝑁) |
| 33 | 22, 32 | jaoi 858 | . 2 ⊢ ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → (𝑁C1) = 𝑁) |
| 34 | 1, 33 | sylbi 217 | 1 ⊢ (𝑁 ∈ ℕ0 → (𝑁C1) = 𝑁) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∨ wo 848 = wceq 1540 ∈ wcel 2108 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 0cc0 11155 1c1 11156 · cmul 11160 < clt 11295 − cmin 11492 / cdiv 11920 ℕcn 12266 ℕ0cn0 12526 ℤcz 12613 ℤ≥cuz 12878 ...cfz 13547 !cfa 14312 Ccbc 14341 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-seq 14043 df-fac 14313 df-bc 14342 |
| This theorem is referenced by: bcnp1n 14353 bcn2m1 14363 bcn2p1 14364 bcnm1 14366 bpoly2 16093 bpoly3 16094 bpoly4 16095 lcmineqlem12 42041 5bc2eq10 42143 jm2.23 43008 |
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